An exercise asks me to find the intersection, sum and product of the ideals generated by $x^2+x-2$ and $x^2-1$ in $\mathbb{Q}[x]$.
The sum is the smaller ideal containing the union so I think I have to find $d = gcd(x^2+x-2,x^2-1)$ and the sum is the ideal $<d>$
For the intersection I have to find the ideal generated by $lcm(x^2+x-2,x^2-1)$ the greater ideal contained in both ideals.
But I don't have clear what should be the product of both ideals. Is it just the product of both polynoms?
The product of two principal ideals is indeed the principal ideal generated by the product of both generators.
By definition, the product of ideals $I$ and $J$ in a ring $R$ is the ideal generated by the set $\{xy|x\in I,y\in J\}$. Thus if $I=(f)$ and $J=(g)$, the product is generated by elements of the form $\lambda fg$ for $\lambda\in R$, which obviously already form an ideal, namely $(fg)$.